Suppose $\mathbf{Z}^n$ is endowed with an associative ring structure so that the coefficient of $e_k$ in $e_i e_j$ is a nonnegative integer for every $i,j,k$. Suppose the ring has an identity element $1$, so that $1 = a_1 e_1 + a_2 e_2 + \cdots + a_n e_n$.
Can any of the $a_i$ ever be negative?
If the $a_i$ are assumed nonnegative, then they are necessarily in $\{0,1\}$. In fact in that case writing $1 = \sum_{s \in S} e_s$, for $S \subset \{1,2,\ldots,n\}$, the $e_s$ must have $e_s^2 = e_s$ and $e_{s'} e_{s''} = 0$ for $s' \neq s''$. I would like to know if this is always true (for a ring with basis with positive integer structure constants)
I think $\Bbb{Z}[x]/(x^2)$ with the basis $\{1+x,x\}$ is a counterexample.